Probability

Probability is the likelihood of an
event
happening. Some other synonyms for probability are chance
and odds. When you say, "I'll
probably be at school tomorrow," you are saying it is likely
you will be at school tomorrow, that there is a good chance you will be
at school tomorrow.

Notation

The notationP(A) is used to show probability. To write that the
probability of rain is 50%, write P(rain)=0.5. Notice
that a decimal number is used, not a percentage.

In probability, we talk about or use:

Event: An occurrence about which probability is measured, calculated, or estimated.

Experiment: An experiment is doing something that makes an event occur.

Probability function: A function that, when given an event, returns an estimated or actual probability of that event occurring. If one adds together the probability of all events, the sum must be 1 since this reflects all possibilities.

Probability distribution graph: A graph that shows how the probability of events is distributed over the sample space.

Example 1: Flipping a Coin

When one flips a coin and records the result, one is doing an
experiment in probability. The parts of the experiment are:

Part

Description

Experiment

The experiment is the whole process of flipping the coin and recording the event.

P(heads) = 0.5,
P(tails)=0.5.
This means that half the time (0.5 = 1/2), the
coin comes up heads and half the time it comes up tails. The sum of all the
possibilities is 0.5 + 0.5 = 1

P(x)

Possible Combinations

Fraction

Decimal

P(H)

1/2

0.5

P(T)

1/2

0.5

Probability Distribution Graph

Figure 1: Probability distribution of a coin flip.

This probability distribution graph
represents
how the probability of each event is distributed over the whole sample space. In this
case the odds of heads is just the same as the odds of tails.

Example 2: Flipping Two Coins

When one flips both coins and records the result, one is doing an
experiment in probability. The parts of the experiment are:

Part

Description

Experiment

The experiment is the whole process of flipping both coins and recording the event. Note that this could also be done by flipping the same coin twice.

Event

The event is flipping both coins.

Outcome

The outcome is one of: both heads, heads then tails, tails then heads, or both tails. This can be abbreviated as HH, HT, TH, and TT where 'H' stands for heads and 'T' stands for tails.

Sample Space

There are 4 outcomes in the sample space. In set notation this is
{HH, HT, TH, TT}.

Probability Function

P(HH)=0.25, P(HT)=0.25,
P(TH)=0.25, P(TT)=0.25. Each of the four outcomes have the same likelihood, or
probability, of occurring. So the probability of each is
1/4 = 0.25. The sum of all the possibilities is
0.25 + 0.25 + 0.25 + 0.25 = 1.0.

P(x)

Possible Combinations

Fraction

Decimal

P(HH)

1/4

0.25

P(HT)

1/4

0.25

P(TH)

1/4

0.25

P(TT)

1/4

0.25

Probability Distribution Graph

Figure 2: Probability distribution of flipping two coins

This probability distribution graph
represents
how the probability of each event is distributed over the whole sample space. In this
case the odds each of the outcomes is equal.

What if one does not care which coin comes up heads and which coin comes up tails? Then
a tails then a heads (TH) is the same as a heads then a tails (HT). This means that there
are 3 outcomes: HH, HT, TT. The parts of the experiment then are:

Part

Description

Experiment

The experiment is the whole process of flipping both coins and recording the event. Note that this could also be done by flipping the same coin twice.

Event

The event is flipping both coins.

Outcome

The outcome is one of: both heads, heads and tails, or both tails. This can be abbreviated as HH, HT, and TT where 'H' stands for heads and 'T' stands for tails.

Sample Space

There are 3 outcomes in the sample space. In set notation this is
{HH, HT, TT}.

Probability Function

P(HH)=0.25, P(HT)=0.5,
P(TT)=0.25. Since there is a one in four chance (P=0.25)
of a heads then a tails and a one in four chance of a tails then a heads, when these
are combined together, we add 0.25 + 0.25 = 0.5. The
sum of all the possibilities is 0.25 + 0.5 + 0.25 = 1.

P(x)

Possible Combinations

Fraction

Decimal

P(HH)

1/4

0.25

P(HT)

or

1/2

0.5

P(TT)

1/4

0.25

Probability Distribution Graph

Figure 3: Probability distribution of flipping 2 coins.

This probability distribution graph
represents
how the probability of each event is distributed over the whole sample space. In this
case the odds each of the outcomes is not equal.

Example 3: Rolling two dice

Now take a look at rolling two dice. Usually one does not care which die has which
value. One only cares about the sum of the dice.

Part

Description

Experiment

The experiment is the whole process of rolling the two dice and recording the event. Note that this could also be done by flipping the same coin twice.

Event

The event is rolling two dice.

Outcome

Since one cares only about the total of the two dice,
the outcome is one of the following:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Sample Space

There are 11 outcomes in the sample space, 2 through 12.
In set notation this is
{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

Probability Function

The probability function for this experiment is:

P(x)

Possible Combinations

Fraction

Approx Decimal

P(2)

1/36

0.0278

P(3)

,

2/36

0.0556

P(4)

,
,

3/36

0.0833

P(5)

,
,
,

4/36

0.1111

P(6)

,
,
,
,

5/36

0.1389

P(7)

,
,
,
,
,

6/36

0.1667

P(8)

,
,
,
,

5/36

0.1389

P(9)

,
,
,

4/36

0.1111

P(10)

,
,

3/36

0.0833

P(11)

,

2/36

0.0556

P(12)

1/36

0.0278

Adding all the probabilities together gives: 36/36 = 1.

Probability Distribution Graph

Figure 4: Probability distribution for rolling 2 six-sided dice.

This probability distribution graph
represents
how the probability of each event is distributed over the whole sample space. In this
case the odds each of the outcomes is not equal.

Principles of Probability

Four basic principles of probability are:

When quantifying, or saying the value, of a probability, we use a number between 0 and 1.
In algebraic notation, for an
arbitrary
event A, 0 ≤ P(A) ≤ 1.

The probability of an impossible event is 0. If
event E is impossible, we write P(E)=0.
This means that the
probability of E occurring is exactly 0. In
real life, very few events are absolutely impossible.

The probability of an event that will certainly happen is 1. If
event H is certain, we write P(H) = 1.

The probability of an event not happening is 1 less the probability
of that event happening. In algebraic notation for an arbitrary event A:
P(!A) = 1 - P(A).
For example, if there is a 30% chance of rain today we write
P(rain) = 0.3. This means that the chance of rain not occurring today is
P(no rain) = 1 - P(rain) = 1 - 0.3 = 0.7.